PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

72 Practical MATLAB® Applications for Engineers

Example 1.12

The set of signals given by

f(t) = ejnwot for n = 0 , ± 1 , ± 2 , ..., ±∞

constitutes an orthogonal family for any arbitrary wo (where wo is referred to as the

fundamental frequency, see Chapter 4 for more details), over the time interval given by

T = 2 π/wo, if the following relation is satisfi ed:

exp( ).exp( )

/

/

/

jnw t jm t dt

wnm

nm

oo

w

w o

o

o



w

 

 

∫







2

0

for

for

Create the MATLAB script fi le ortog that verifi es the preceding relation for the following

arbitrary values of wo , n, and m:

a. wo = 2 , n = 5 , and m = 5

b. wo = 2 , n = 5 , and m = 6

c. wo = 7 , n = 8 , and m = 10

d. wo = 7 , n = 10 , and m = 10

MATLAB Solution

% Script file: ortog

syms check _ a check _ b check _ c check _ d expon t ;

expon = exp(j*5*2*t)*exp(-j*5*2*t);%wo=2,n=5,m=5

check _ a = int(expon,-pi/2,pi/2);

% f = vpa(check _ a)

expon = exp(j*5*2*t)*exp(-j*6*2*t);%wo=2,n=5,m=6

check _ b = int(expon,-pi/2,pi/2);

expon = exp(j*8*7*t)*exp(-j*10*7*t);%wo=7,n=8,m=10

check _ c = int(expon,-pi/7,pi/7);

expon = exp(j*10*7*t)*exp(-j*10*7*t);%wo=7,n=10,m=10

check _ d = int(expon,-pi/7,pi/7);

disp(‘****************RESULTS***********************************’)

disp(‘The results for parts (a),(b),(c) and (d) are given by :’)

results = [check _ a check _ b check _ c check _ d];

disp(results)

disp(‘***********************************************************’)

Back in the command window the script fi le ortog is executed and the results are

shown as follows:

>> ortog

************************* RESULTS ****************************

The results for parts (a), (b), (c) and (d) are given by:

[ pi, 0, 0, 2/7*pi ]

***************************************************************

Observe that the results obtained clearly confi rm that the exponential family of func-

tions, given by f(t) = ejnwot, for n = 0 , ± 1 , ± 2 , ..., ±∞ constitute an orthogonal family.